Problem 18 - Applied Calculus

Question

(a) What are the equilibrium solutions for the differential equation $$ \frac{d y}{d t}=0.2(y-3)(y+2) ? $$ (b) Use a graphing calculator or computer to sketch a slope field for this differential equation. Use the slope field to determine whether each equilibrium solution is stable or unstable.

Step-by-Step Solution

Answer

Equilibrium solutions are \( y = 3 \) (unstable) and \( y = -2 \) (stable).

Step 1: Identify Equilibrium Solutions

To find equilibrium solutions, we set \( \frac{dy}{dt} = 0 \). This means the expression \( 0.2(y-3)(y+2) = 0 \) must equal zero. For the product to be zero, either \( y-3 = 0 \) or \( y+2 = 0 \). Solving, we find the equilibrium solutions are \( y = 3 \) and \( y = -2 \).

Step 2: Sketch the Slope Field

For sketching the slope field, analyze the behavior of \( \frac{dy}{dt} = 0.2(y-3)(y+2) \) around the equilibrium points. At \( y = 3 \) and \( y = -2 \), the slope (rate of change of \( y \)) is zero. Use a graphing calculator to plot \( \frac{dy}{dt} \) at several points to visualize how the slope behaves around these equilibria.

Step 3: Determine Stability from Slope Field

Examine the slope field around each equilibrium. For \( y = 3 \), if solutions above 3 move toward 3 and below 3 move away, it is unstable. For \( y = -2 \), if solutions above and below move towards \(-2\), it is stable.

Key Concepts

Differential Equation

A differential equation is an equation that involves derivatives of a function and provides a way to describe change. In our example, the differential equation is \( \frac{dy}{dt} = 0.2(y-3)(y+2) \).
This equation governs how the variable \( y \) changes with respect to time \( t \).
Understanding this concept is crucial in fields ranging from physics to biology, as it helps model real-world phenomena like population growth and chemical reactions.

• Variable \( y \) is often called the dependent variable.
• Derivative \( \frac{dy}{dt} \) indicates the rate of change of \( y \) over time.
• The equation can be complex or simple, linear or nonlinear based on the relationship between \( y \) and its derivatives.

By solving differential equations, we find particular solutions or behaviors of the system over time.
Here, finding the equilibrium solutions involves setting the derivative equal to zero to describe where changes cease.

Slope Field

A slope field, also known as a direction field, provides a graphical representation of a differential equation. It illustrates how slopes, or instantaneous rates of change, behave at different points in the plane.
In our case, the slope field for the equation \( \frac{dy}{dt} = 0.2(y-3)(y+2) \) shows the dynamics around our equilibrium solutions.

• Each point in the slope field has a small line segment with a slope corresponding to the value of the derivative at that point.
• By observing the slope field, students can visualize how solutions to the differential equation behave over time.
• This tool is especially useful when exact solutions are hard to obtain analytically.

Using a graphing calculator can be particularly helpful in generating slope fields, allowing for a deeper understanding of the equation's behavior without extensive manual calculations.

Stability Analysis

Stability analysis involves evaluating equilibrium solutions to understand if small perturbations will cause a system to return to equilibrium (stable) or deviate further (unstable).
In our problem, we identified two equilibrium points: \( y = 3 \) and \( y = -2 \).

• An equilibrium solution is considered stable if trajectories near it tend to stay near or return to that equilibrium.
• It is unstable if likely small deviations lead to trajectories moving further away.

For \( y = 3 \), the surrounding slopes suggest it's potentially an unstable equilibrium since trajectories below \( y = 3 \) move away, not returning.
Conversely, at \( y = -2 \), slopes around this point attract solutions towards it, indicating stability.
The slope field assists in this analysis by visually demonstrating the flow of solutions.

Graphing Calculator

Using a graphing calculator can significantly aid in understanding differential equations, especially for sketching slope fields and analyzing stability.
Here's how it helps:

• Graphing calculators can quickly plot the slope field, providing a visual map of solution behaviors.
• They allow for zooming in and out to focus on specific areas, such as close examination of equilibrium points.
• With built-in tools, they can qualitatively determine stability by simulating trajectories around equilibrium points.

While manually computing these slopes can be tedious, graphing calculators offer accurate and faster alternatives.
Integrating this technology into solving differential equations complements analytical skills, offering students hands-on visual experience.